Find dy/dx given x = t² - 2t and y = t⁴ - 4t?

Solution:

We apply differentiation of parametric functions as x = f(t) and y = g(t)

Let us find dy/dx by finding dy/dt and dx/dt

dy/dx = dy/dt . dt/dx

x = t² - 2t

dx/dt = 2t - 2

y = t⁴ - 4t

dy/dt = 4t³ - 4

dy/dx = dy/dt / dx/dt

= ( 4t³ - 4)/(2t - 2)

= 4(t³ - 1)/2(t - 1)

=   2(t³ - 1)/(t - 1)

We also know that a³ - b³ = (a - b)(a² + ab + b²) 

Therefore t³ - 1 = t³ - (1)³  = (t - 1)(t²  + t + 1)

dy/dx = 2(t - 1)(t²  + t + 1)/ (t - 1)

= 2(t²  + t + 1)

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Find dy/dx given x = t² - 2t and y = t⁴ - 4t?

Summary:
 dy/dx given x = t² - 2t and y = t⁴ - 4t is  2(t²  + t + 1)